A174455 Number of partitions where the number of 1's and 2's are equal.

1, 0, 0, 2, 1, 1, 4, 3, 4, 8, 8, 10, 17, 18, 23, 34, 39, 48, 67, 78, 97, 127, 151, 185, 237, 281, 343, 428, 511, 616, 759, 902, 1084, 1315, 1562, 1863, 2242, 2649, 3147, 3752, 4424, 5222, 6190, 7266, 8545, 10062, 11776, 13782, 16157, 18832, 21964, 25622, 29777, 34589, 40200, 46556, 53912

Offset: 0

Author

Joerg Arndt, Nov 28 2010

Keywords

• nonn

Comments

From Omar E. Pol, Jan 19 2013: (Start)

Column 3 of triangle A220504.

With offset 3, a(n) is also the number of appearances of 3 as the smallest part in all partitions of n.

Also consider the sequence formed by [0, 0] together with this sequence, with offset 1, then it appears that A027336(n) = Sum_{j=1..3} a(n+j), n >= 0.

(End)

Examples

			a(8)=9, there are 8 such partitions of 9, they are
  #1:    9 =  3* 1 + 3* 2 + 0    + 0    + 0    + 0    + 0    + 0    + 0
  #2:    9 =  2* 1 + 2* 2 + 1* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #3:    9 =  1* 1 + 1* 2 + 2* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #4:    9 =  0    + 0    + 3* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #5:    9 =  0    + 0    + 0    + 1* 4 + 1* 5 + 0    + 0    + 0    + 0
  #6:    9 =  1* 1 + 1* 2 + 0    + 0    + 0    + 1* 6 + 0    + 0    + 0
  #7:    9 =  0    + 0    + 1* 3 + 0    + 0    + 1* 6 + 0    + 0    + 0
  #8:    9 =  0    + 0    + 0    + 0    + 0    + 0    + 0    + 0    + 1* 9
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Cf. A008483, A182713, A240056.

Programs

Maple

b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
      else `if`(i=3 and irem(n, 3, 'r')=0, r, 0); for j from 0
      to n/i do %+b(n-i*j, i-1) od; % fi
    end:
a:= n-> b(n+3, n+3):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 20 2013

Mathematica

(* See A240056. - Clark Kimberling, Mar 31 2014 *)
m = 66; gf = 1/((1-x^3)*Product[1-x^n, {n, 3, m}]) + O[x]^m; CoefficientList[gf, x] (* Jean-François Alcover, Jul 02 2015, after Joerg Arndt *)

PARI

N=66;  x='x+O('x^N);
gf=1/( (1-x^3) * prod(n=3,N, 1-x^n) );
Vec(gf)
/* Joerg Arndt, Jul 07 2012 */

Formula

G.f.: 1/( (1-x^3) * Product_{n>=3} (1-x^n) ). - Joerg Arndt, Jul 07 2012

a(n) = A182713(n+2) - A182713(n) = A240056(n+1) - A240056(n) for n >= 0. - Clark Kimberling, Mar 31 2014

a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (9 * 2^(3/2) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2022